of Force between Electric Currents. 459 



law of action between current-elements is necessarily such 

 that the resultant action of a closed circuit on an element is 

 normal to that element ; they would be equally satisfied by any 

 law of action giving a potential between two closed circuits. 



14. Indeed it may be shown directly that, whether there be 

 any tangential action or no, it never can be manifested by 

 experimenting upon a closed circuit ; and therefore, if all cir- 

 cuits be necessarily closed circuits, it follows that such tan- 

 gential action never can be manifested. 



For such closed circuit must be infinitely flexible and infi- 

 nitely extensible (at least in the neighbourhood of the element 

 acted on), otherwise the element will be influenced by the 

 mechanical action of the adjacent parts of its own circuit; and 

 if it be perfectly flexible and extensible, no virtual motion of 

 any element in the direction of its length can alter the value 

 of the potential 



cose , 



asas . 



s- 



15. Again, it is generally stated to be a deduction from 

 Ampere's law, that the action of a solenoid of currents of in- 

 nitely small section and indefinitely extended in one direction, 

 upon an element of a current, is a force perpendicular to the 

 plane passing through the element and the extremity of the 

 solenoid, varying inversely as the square of the distance of the 

 element from that extremity, and directly as the sine of the 

 angle between the element and that distance. Ampere's law 

 would give exactly this result, while that now proposed would 

 not do so, for an element considered alone, i. e. otherwise than 

 as part of a closed or infinite current. It is clear, for the 

 reasons mentioned above, that no experiment can give a result 

 in this form. What experiment really proves is, that in the 

 case of a closed circuit or infinitely extended rectilinear cur- 

 rent, the action on the solenoid is the same as if that of each 

 element of the current followed the above law. And the law 

 of F. E. Neumann leads exactly to the same conclusion. For 

 let the infinite current be of strength i in the axis of z, z ; then 

 it is evident by integration that Ampere's law of elementary 

 current action would lead to a potential energy between the 



whole current and the solenoid of the form ^tan~ J -, or, more 



(X? 7 



generally, when the solenoid is not infinite, to the form 



i< tan - 1 ^ — tan -1 - !-. 

 (. x a J 



For according to the law now proposed, the potential between 



